Modelling the compaction of plastic particle packings

Abstract

Soft particle materials such as some pharmaceutical and food products are composed of particles that can undergo large deformations under low confining pressures without rupture. The rheological and textural properties of these materials are thus governed by both particle rearrangements and particle shape changes. For the simulation of soft particle materials, we present a numerical technique based on the material point method, allowing for large elasto-plastic particle deformations. Coupling the latter with the contact dynamics method makes it possible to deal with contact interactions between particles. We investigate the compaction of assemblies of elastic and plastic particles. For plastic deformations, it is observed that the applied stress needed to achieve high packing fraction is lower when plastic hardening is small. Moreover, predictive models, relating stress and packing fraction, are proposed for the compaction of elastic and plastic particles. These models fit well our simulation results. Furthermore, it is found that the evolution of the coordination number follows a power law as a function of the packing fraction beyond jamming point of hard particle packings.

Appendix 1: Relation between the applied stress, \(\sigma \), and the packing fraction, \(\varPhi \), for an elastic packing under uniaxial compression

We assume that the packing of particles behaves almost as a continuum porous medium beyond the jamming point under uniaxial compression. Hence, in this range the applied stress $\sigma$ may be related to the cumulative vertical strain $\varepsilon$ through an effective P-wave modulus $M^{\mathrm{eff}}$:

$$\begin{aligned} \begin{array}{c} \displaystyle { \sigma = M^{\mathrm{eff}} \varepsilon \ . } \end{array} \end{aligned}$$

(18)

One may further assume that the particle bulk modulus K relates the volume increment $\text {d} V_S$ of particles to the effective stress increment $\text {d} \sigma _S$ in particles:

$$\begin{aligned} \begin{array}{c} \displaystyle { K \frac{\text {d} V_S}{V_S} = -\text {d} \sigma _S \ . } \end{array} \end{aligned}$$

(19)

$\sigma$ can be related to $\sigma _S$ as follows:

$$\begin{aligned} \begin{array}{c} \displaystyle { \sigma = c_1 Z \varPhi \sigma _S \ , } \end{array} \end{aligned}$$

(20)

where $c_1$ is a material constant to be determined. Given that $\text {d} \varepsilon =\text {d} V_S/V_S-\text {d} \varPhi / \varPhi$ and using Eqs. (18), (19) and (20), the following differential equation to solve is obtained:

$$\begin{aligned}&\left( 1+ \frac{M^{\mathrm{eff}} }{c_1 K Z \varPhi }\right) \text {d} \sigma = \left[ \frac{M^{\mathrm{eff}} }{c_1 K Z \varPhi } \left( \frac{\text {d}Z}{Z} + \frac{{\text {d}\varPhi }}{{\varPhi }}\right) + \frac{\text {d}M^{\mathrm{eff}} }{M^{\mathrm{eff}} } \right] \sigma \nonumber \\&\qquad - M^{\mathrm{eff}} \frac{ {\text {d} \varPhi }}{{\varPhi }}. \end{aligned}$$

(21)

By knowing that $M^{\mathrm{eff}}$ and Z are related to $\varPhi$ (see Eqs. (7) and (17)), the integration of the differential equation (21) yields:

$$\begin{aligned} \begin{array}{c} \displaystyle { \sigma = - \frac{M^{\mathrm{eff}}}{1+\frac{M^{\mathrm{eff}}}{c_1K Z \varPhi }} (\ln (\varPhi ) + c_2) \ , } \end{array} \end{aligned}$$

(22)

where $c_2$ is the integral constant.